Why is the weierstrass function pathological?

Question

The weierstrass function is one of famous examples of pathological function. The property of that function is continuous for all real numbers, but not differentiable everywhere.

Before discovering the pathological function, every continuous function was considered to be differentiable.

I do not understand the thought that The continuous functions are differentiable. So,I do not know why the weierstrass function is pathological. Also,I would like to gain views of the weierstrass function from present mathematics.

I wish you answer this question.

Answer 1

I think the view of the Weierstrass function nowadays is

1. It's not pathological – it's typical – there is a well-defined sense in which most everywhere continuous functions are nowhere differentiable,

2. It has the advantage over other functions with the above properties of being easy enough to describe that even an undergraduate can understand it, so it's handy when, for whatever reason, you need a function with those properties,

3. Other than that, no one really cares about it very much. I mean, there's a million dollars on offer for figuring out what the Riemann zeta function does; there are no prizes for settling open problems about the Weierstrass function.